Quaternary Science Reviews 26 (2007) 37–55 Glacial variability over the last two million years: an extended depth-derived agemodel, continuous obliquity pacing, and the Pleistocene progression Peter Huybers Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods Hole, MA, 02540, USA Received 11 November 2005; received in revised form 21 May 2006; accepted 21 July 2006 Abstract An agemodel not relying upon orbital assumptions is estimated over the last 2 Ma using depth in marine sediment cores as a proxy for time. Agemodel uncertainty averages 10 Ka in the early Pleistocene (2–1 Ma) and 7 Ka in the late Pleistocene (1 Ma to the present). Twelve benthic and five planktic d 18 O records are pinned to the agemodel and averaged together to provide a record of glacial variability. Major deglaciation features are identified over the last 2 Ma and a remarkable 33 out of 36 occur when Earth’s obliquity is anomalously large. During the early Pleistocene deglaciations occur nearly every obliquity cycle giving a 40 Ka timescale, while late Pleistocene deglaciations more often skip one or two obliquity beats, corresponding to 80 or 120 Ka glacial cycles which, on average, give the 100 Ka variability. This continuous obliquity pacing indicates that the glacial theory can be simplified. An explanation for the 100 Ka glacial cycles only requires a change in the likelihood of skipping an obliquity cycle, rather than new sources of long-period variability. Furthermore, changes in glacial variability are not marked by any single transition so much as they exhibit a steady progression over the entire Pleistocene. The mean, variance, skewness, and timescale associated with the glacial cycles all exhibit an approximately linear trend over the last 2 Ma. A simple model having an obliquity modulated threshold and only three adjustable parameters is shown to reproduce the trends, timing, and spectral evolution associated with the Pleistocene glacial variability. r 2006 Elsevier Ltd. All rights reserved. Keywords: Glacial cycles; Mid-Pleistocene transition; Geochronology; Obliquity; Orbital forcing; Hypothesis test 1. Introduction The onset of glaciation near 3 Ma is thought to owe to a gradual cooling trend over the last 4 Ma (Shackleton and Hall, 1984; Raymo, 1994; Ravelo et al., 2004) which is itself part of a longer-term trend over the last 50 Ma (Zachos et al., 2001). Early-Pleistocene (2–1 Ma) glacial cycles have a 40 Ka timescale; thus these cycles are readily attributed to the 40 Ka changes in Earth’s obliquity (e.g. Raymo and Nisancioglu, 2003; Huybers, 2006). In contrast, late-Pleistocene (1 Ma-present) glacial cycles have a longer 100 Ka timescale often attributed to orbital precession (Hays et al., 1976; Imbrie et al., 1992; Ghil, 1994). Existing hypotheses for this ‘‘mid-Pleistocene transition’’ (or mid-Pleistocene revolution) from 40 to 100 Ka glacial cycles call for shifts in the controls on glaciation to activate new sources of low-frequency variability (Saltzman and Sutera, 1987; Maasch and Saltzman, 1990; Ghil, 1994; Raymo, 1997; Paillard, 1998; Clark et al., 1999; Tziperman and Gildor, 2003; Ashkenazy and Tziperman, 2004). The recent results of Huybers and Wunsch (2005) (hereafter HW05), however, show that the late Pleistocene glacial terminations are paced by changes in Earth’s obliquity, suggesting that a more unified glacial theory is possible, ARTICLE IN PRESS 0277-3791/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.quascirev.2006.07.013 Harvard University, Earth and Planetary Sciences, Museum Bldg. rm405, 20 Oxford St., Cambridge MA 02138, USA. Tel.: +1 617 233 3295; fax: +1 508 457 2187. E-mail addresses: [email protected], [email protected].
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Quaternary Science Reviews 26 (2007) 37–55
Glacial variability over the last two million years: an extendeddepth-derived agemodel, continuous obliquity pacing, and the
Pleistocene progression
Peter Huybers�
Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods Hole, MA, 02540, USA
Received 11 November 2005; received in revised form 21 May 2006; accepted 21 July 2006
Abstract
An agemodel not relying upon orbital assumptions is estimated over the last 2Ma using depth in marine sediment cores as a proxy for
time. Agemodel uncertainty averages �10Ka in the early Pleistocene (�2–1Ma) and �7Ka in the late Pleistocene (�1Ma to
the present). Twelve benthic and five planktic d18 O records are pinned to the agemodel and averaged together to provide a record of
glacial variability. Major deglaciation features are identified over the last 2Ma and a remarkable 33 out of 36 occur when Earth’s
obliquity is anomalously large. During the early Pleistocene deglaciations occur nearly every obliquity cycle giving a 40Ka timescale,
while late Pleistocene deglaciations more often skip one or two obliquity beats, corresponding to 80 or 120Ka glacial cycles which,
on average, give the �100Ka variability. This continuous obliquity pacing indicates that the glacial theory can be simplified. An
explanation for the �100Ka glacial cycles only requires a change in the likelihood of skipping an obliquity cycle, rather than new sources
of long-period variability. Furthermore, changes in glacial variability are not marked by any single transition so much as they exhibit
a steady progression over the entire Pleistocene. The mean, variance, skewness, and timescale associated with the glacial cycles all
exhibit an approximately linear trend over the last 2Ma. A simple model having an obliquity modulated threshold and only
three adjustable parameters is shown to reproduce the trends, timing, and spectral evolution associated with the Pleistocene glacial
variability.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Glacial cycles; Mid-Pleistocene transition; Geochronology; Obliquity; Orbital forcing; Hypothesis test
1. Introduction
The onset of glaciation near 3Ma is thought to owe toa gradual cooling trend over the last 4Ma (Shackletonand Hall, 1984; Raymo, 1994; Ravelo et al., 2004) which isitself part of a longer-term trend over the last 50Ma(Zachos et al., 2001). Early-Pleistocene (�2–1Ma) glacialcycles have a 40Ka timescale; thus these cycles arereadily attributed to the 40Ka changes in Earth’s obliquity(e.g. Raymo and Nisancioglu, 2003; Huybers, 2006). In
e front matter r 2006 Elsevier Ltd. All rights reserved.
ascirev.2006.07.013
niversity, Earth and Planetary Sciences, Museum Bldg.
rd St., Cambridge MA 02138, USA. Tel.: +1 617 233 3295;
contrast, late-Pleistocene (�1Ma-present) glacial cycleshave a longer �100Ka timescale often attributed to orbitalprecession (Hays et al., 1976; Imbrie et al., 1992; Ghil,1994).Existing hypotheses for this ‘‘mid-Pleistocene transition’’
(or mid-Pleistocene revolution) from 40 to �100Ka glacialcycles call for shifts in the controls on glaciation to activatenew sources of low-frequency variability (Saltzman andSutera, 1987; Maasch and Saltzman, 1990; Ghil, 1994;Raymo, 1997; Paillard, 1998; Clark et al., 1999; Tzipermanand Gildor, 2003; Ashkenazy and Tziperman, 2004). Therecent results of Huybers and Wunsch (2005) (hereafterHW05), however, show that the late Pleistocene glacialterminations are paced by changes in Earth’s obliquity,suggesting that a more unified glacial theory is possible,
related to obliquity both during the early and latePleistocene. Here, the argument is made that the progres-sion from 40 to �100Ka glacial cycles is not marked byany specific transition and does not require any real changein the physics of glacial cycles.
The paper is organized as follows: In Section 2 the depth-derived agemodel of Huybers and Wunsch (2004) (here-after HW04) is extended from 0.8 to 2Ma. The reader notinterested in agemodel construction may safely skip thissection. In Section 3 the hypothesis testing proceduresoutlined in HW05 (but now applied to many moreobservations) are used to evaluate the relationship betweenorbital variations and glacial variability. Section 4 discussesthe trends in Pleistocene glacial variability, and Section 5presents a simple model which describes these trends andthe timing of the glacial cycles. The paper is concluded inSection 6.
2. The timing of Pleistocene glaciation
The use of age estimates not depending on orbitalassumptions are required to avoid circular reasoning whenassessing the link between glacial and orbital variability.This section presents an extension of the depth-derivedagemodel of HW04 from 0.8 to 2.0Ma. This agemodelimproves on previous non-orbitally tuned Pleistocene ageestimates (Williams et al., 1988; Raymo and Nisancioglu,2003) by combining numerous age–depth relationships,correcting for down-core compaction, and by providing forage uncertainty estimates. Most agemodels spanning thePleistocene (e.g. Shackleton et al., 1990; Lisiecki andRaymo, 2005) constrain ages by aligning variations in thed18 O record with variations in the orbital parameters, thusprecluding an objective evaluation of the orbital influenceon glacial timing.
Table 1
Sediment cores
Name Reference
DSDP607 Ruddiman et al. (1989)
MD900963 Bassinot et al. (1994)
ODP663 de Menocal et al. (unpublished)
ODP664 Raymo (1997)
ODP677 Shackleton et al. (1990)
ODP846 Mix et al. (1995a)
ODP849 Mix et al. (1995b)
ODP925 Bickert et al. (1997) and Curry and Cullen (1997)
ODP927 Cullen and Curry (1997), and Curry and Cullen (1997)
ODP980 Flower (1999), McManus et al. (1999, 2002), and Oppo et
ODP982 Venz et al. (1999).
ODP983 Channell et al. (1997) and McManus et al. (2003)
TT013-PC18 Murray et al. (2000)
TT013-PC72 Murray et al. (2000)
Characteristics and primary references for each core. Columns from left to righ
accumulation rate (S, cm/Ka), the mean interval between d18 O measurements
2.1. Converting depth to age
Conversion of sediment core depths into age estimates isbased on the graphic correlation methodology of Shaw(1964) and detailed in HW04. Only those instances wherethe methodology is extended or altered from HW04 aredwelled on here. Geomagnetic age control comes from theBrunhes–Matuyama transition (B–M) (0.78Ma), theJaramillo (0.99–1.07Ma), and Olduvai (1.77–1.95Ma)subchrons, and the Matuyama–Gauss transition(2.58Ma) (Berggren et al., 1995; Cande and Kent, 1995).These geomagnetic ages are derived from a combination ofradiometric dating techniques, sea-floor spreading rates,and astronomically derived age estimates—the last indicat-ing that geomagnetic age estimates do contain limitedorbital assumptions. However, only six geomagnetic agesare utilized in the course of 2Ma and the orbitalassumptions are but one constraint on the geomagneticages, minimizing the influence of the orbital assumptions.Also, the B–M is radiometrically dated to sufficientaccuracy ð�2KaÞ (Singer and Pringle, 1996) that it isessentially independent of orbital assumptions, and thus sois the agemodel between the present and 0.78Ma.Geomagnetic events are assumed to occur in the sameisotopic stages identified by Raymo and Nisancioglu (2003)for core DSDP607.References and statistics for each sediment core used in
this study are given in Table 1. Approximately synchro-nous time horizons are identified between sediment coresby identifying corresponding d18 O events in the separatestratigraphies. (Event synchronization is limited by thesignal propagation time through the ocean and theresolution of the d18 O stratigraphy.) HW04 identified 17events over 780Ka, giving an average event spacing of46Ka. Here, a total of 104 d18 O events are identifiedbetween the present and the Matuyama–Gauss transitionproviding an average d18 O event spacing of 24Ka (see
Species S nt Lat. Lon.
B 4.0 3.5 41N 33W
P 4.6 2.3 5N 74E
P 3.9 3.0 1S 12W
B 3.7 3.4 0 23W
B,P 3.9 2.1,1.8 1N 84W
B 3.7 2.5 3S 91W
B 2.9 3.6 0 111W
B 3.7 2.2 4N 43W
B,P 4.5 3.2,2.2 6N 43W
al. (1998, 2001). B 12.3 1.6 55N 17W
B,P 2.5 2.3,2.0 57N 18W
B 11.4 0.9 61N 22W
B 1.5 3.7 2S 140W
B 1.6 3.3 0 139W
t display d18 O species benthic (B) and/or planktic (P), the mean sediment
ðnt;KaÞ, and the latitude and longitude of each core site.
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Fig. 1. Planktic d18 O records pinned to the extended depth-derived agemodel. Note that time runs from left to right. Shown are anomalies with respect to
0.7–0Ma, records are offset from one another by 1:75%. Vertical lines indicate geomagnetic events used for age control (not showing the
Matuyama–Gauss reversal at 2.58Ma). Dots indicate events pinned to the depth-derived agemodel. Vertical shading indicates where more than one
obliquity cycle elapses between deglacial events. At bottom is the averaged planktic stack with age-control points labeled according to the isotopic stage
notation of Ruddiman et al. (1989).
P. Huybers / Quaternary Science Reviews 26 (2007) 37–55 39
Figs. 1 and 2). Event identification follows the d18 O stagenotation of Ruddiman et al. (1989), Raymo et al. (1989),and Shackleton (1995). Increasing the number of age-control points decreases agemodel uncertainty and betterpreserves the structure of the d18 O variability when recordsare averaged (Huybers, 2004).
In order to accurately identify d18 O features at aresolution of 24Ka, a relatively high d18 O sampling rateis required. Only those records having a sampling resolu-tion of four Ka or less are included in this study, reducingthe total number of records from 21 in HW04 to 14 here.This leads to small changes in the depth-derived ages of�4Ka, consistent with the estimated uncertainties. Detailsof this agemodel and differences with respect to that ofHW04 are given in the supplementary material.
Ages are assigned to each d18 O event by interpolatingage with depth between the geomagnetic events. Thisprovides as many estimates of the age of each d18 O event asthere are sediment cores. The averages of the event ages areexpected to be more accurate than any single event age andserve as a master chronology. A complete agemodel isestimated for each sediment core by linearly interpolatingage with depth between each of the average event ages.Note that following the methodology of HW04, depth ineach sediment core has been corrected for the effects ofcompaction.
There are 14 sediment cores which extend to the B–Mtransition, nine to the bottom of the Jaramillo, four to thebottom of the Olduvai, and two extend back to theMatuyama–Gauss transition. The choice is made totruncate cores at the oldest geomagnetic event which theyreach. This prevents having to extrapolate age–depthrelationships outside of regions bounded by geomagneticor core-top age control. Such a truncation is chosenbecause age error is expected to grow six times morequickly when one end of the age–depth relationship is notconstrained in time (HW04).
2.2. Agemodel uncertainty
A Monte Carlo approach is used to estimate agemodeluncertainty. Uncertainty estimates account for sedimentaccumulation rate variations, sediment compaction, thetime for the d18 O signal to propagate throughout theoceans, and uncertainty in magnetic event ages and theiridentification relative to d18 O stages. Unless otherwisestated, uncertainty estimates follow HW04.It is assumed that there are as many degrees of freedom
as there are sediment cores. This differs from HW04 wherethe degrees of freedom were assumed to be fewer than thetotal number of cores owing to covariation betweenaccumulation rates at different cites. Further analysis,
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Fig. 2. Similar to Fig. 3 but for benthic d18 O records.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–5540
based on the cores used in this study, indicates thataccumulation rate variations are as likely to be anti-correlated as they are to be positively correlated so thatthere appears no reason to reduce the total number ofdegrees of freedom. This assumption of greater degrees offreedom along with a greater number of d18 O eventsreduces average uncertainty estimates from �9Ka inHW04 to �7Ka for the period between the B–M and thepresent.
The correction for sediment compaction, on average,makes ages 10Ka younger between the B–M and thepresent. The influence of compaction on earlier ages ismuch smaller—on average making ages 1Ka younger—primarily because changes in compaction are less pro-nounced further down-core (Bahr et al., 2001). Asecondary reason is that geomagnetic events, which fixevent ages independent of compaction, are more closely
spaced between the B–M and Matuyama–Gauss transi-tions than between the B–M and present. The averagemagnitude of the compaction correction uncertainty belowthe B–M is taken to be �0:5 ka.Geomagnetic ages are assumed to be known to within�5Ka (Berggren et al., 1995; Cande and Kent, 1995)except for the B–M which is known to within �2Ka. Anadditional �2Ka is added to account for uncertainty in theidentification of when a geomagnetic event occurs within asediment core (Tauxe et al., 1996).The estimated agemodel uncertainty is shown in Fig. 3.
Uncertainty tends to grow away from geomagnetic andd18 O events and follows a Brownian bridge structure (seeHW04). Agemodel uncertainty averages �10Ka during theearly Pleistocene and �7Ka during the late Pleistocene.Late-Pleistocene ages are more certain because of thegreater number of sediment cores.
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Fig. 3. Estimated agemodel uncertainty. Shown is the one standard deviation of 104 Monte Carlo agemodel realizations. Dots indicate the isotopic events
used to align the records and vertical bars indicate geomagnetic age constraints, excepting the youngest bar which is the radiometrically dated termination
one feature.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–55 41
Although not relying on orbital assumptions itself, theextended depth-derived agemodel is similar to orbitallytuned estimates. The difference between the extendeddepth-derived ages and the orbitally tuned agemodelof Lisiecki and Raymo (2005) has a standard deviationðsÞ of only �6Ka, less than the expected uncertaintyfor the depth-derived ages. The small standard devi-ation could owe to the fact that the Lisiecki and Raymo(2005) agemodel estimation procedure explicitly seeksto minimize variations in accumulation rate, an approachnot unlike the depth-derived procedure. Differencesbetween the extended depth-derived agemodel and theagemodel of Shackleton (1995) have s ¼ �7Ka. Interest-ingly, the two orbitally tuned agemodels (Shackleton, 1995;Lisiecki and Raymo, 2005) have discrepancies over thelast 2Ma of s ¼ �6Ka, similar to the discrepanciesbetween the depth-derived and orbitally tuned age esti-mates. It can be inferred that the orbitally tuned ageestimates are unlikely to be more accurate than the depth-derived ages.
2.3. Averaged d18 O record
To reduce noise owing to measurement error and localclimate variability (e.g. Mix, 1987), the individual d18 Orecords are averaged using the extended depth-derivedagemodel. Prior to averaging, however, it is necessary toaccount for mean offsets between the d18 O records, owingprimarily to mean differences in water temperature but alsoto differences in the ambient d18 O of the water and vitaleffects (e.g. Lynch-Stieglitz et al., 1999). Otherwise, whenshort records drop out back in time the mean value wouldchange. To account for these mean offsets in d18 O, therecord average between 0.7Ma and the present issubtracted from each record.
Averaging requires that all records first be interpolatedto the same sample spacing. To account for differingsampling resolution between cores, records are weightedaccording to the inverse of the average sampling resolution,
yðtÞ ¼XNðtÞ
i¼1
wðt; iÞ � y0ðt; iÞ,
wðt; iÞ ¼ uðtÞ � sðiÞ�1. ð1Þ
Here, yðtÞ is the average d18 O at time t, NðtÞ is the numberof records available as, function of time, and y0ðt; iÞ is thed18 O anomaly of the ith record relative to the averagebetween 0.7Ma and the present. The average samplingresolution is given by sðiÞ and u is a constant chosen sothat the sum of the weights always equals unity,PNðtÞ
i¼1 wðt; iÞ ¼ 1. Eq. (2) ensures that the contribution ofeach record to the average is proportional to the number ofdata points it contributes. Prior to averaging, records aresmoothed using a running 5Ka average to help suppressnoise and local climate variability.Fig. 4 shows the averaged d18 O record, referred to as the
stack, plotted against the d18 O compilation of Lisiecki andRaymo (2005). The two records have very similarstructures. When the Lisiecki and Raymo (2005) compila-tion is placed on the depth-derived agemodel (see thesupplementary material) the cross correlation with thedepth-derived stack is 0.95.An evolutionary spectrum of the depth-derived stack
shows the usual features: variability is primarily concen-trated at 40Ka periods during the early Pleistocene and at�100Ka periods during the late Pleistocene. The onset of100Ka variability is concomitant with increased variabilitynear �20Ka. Prior to drawing any conclusion from theevolutionary spectrum, however, it is useful to analyze thestack using a few other statistical methods.
3. A test of the orbital hypothesis
HW05 showed that the timing of glacial terminationsduring the late Pleistocene coincide with periods ofincreased obliquity. Here, those results are extended toinclude both early-Pleistocene deglacial events and smalleramplitude deglacial events during the late Pleistocene.The increased number of observations increases thestatistical power of the test. Furthermore, the longerrecord places the late-Pleistocene glacial termination inthe perspective of the earlier, smaller amplitude, andshorter period variations.A formal hypothesis test is conducted. The hypothesis,
H1, is that deglaciations are triggered at a particularphase of Earth’s obliquity. The null hypothesis, H0, isthat deglaciations are independent of the phase ofobliquity. Discussion is framed around obliquity because
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Fig. 4. Glacial variability over the last 2Ma. The averaged d18 O record is shown at top on the depth-derived agemodel (thick line). For comparison, the
d18 O compilation of Lisiecki and Raymo (2005) is also shown (thin line). Units are in % and the mean between 700Ka and the present has been removed.
An evolutionary spectrum of the depth-derived record is shown at bottom. Shading indicates the log10 of the spectral power. Spectra are calculated using a
400Ka sliding window. Horizontal dashed lines are at 1/100, 1/41, and 1/22Ka.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–5542
the early-Pleistocene variability is characterized by 40Kaperiods (Raymo and Nisancioglu, 2003) and the late-Pleistocene glacial terminations are known to initiateduring times of increased obliquity (HW05). Precessionand eccentricity are also considered. The orbital pacing ofthe early (�2–1Ma) and late-Pleistocene (1Ma to thepresent) glacial variability are tested separately as theseare generally characterized as distinct modes of glacialvariability.
To conduct the hypothesis test three elements arerequired: (1) an objective identification of what constitutesa deglacial event and when it occurs, (2) a test statistic tomeasure the stability of deglacial timing with respect toorbital variations, and (3) an estimate of the probabilitydistribution functions (PDFs) associated with H0 and H1.Each element is discussed in turn.
3.1. Identification
The criterion adopted for identification of deglacialevents is that the increase in d18 O (decrease in ice volume)between a local minimum and the following maximummust exceed one standard deviation of the d18 O record, i.e.greater than 0:35%. To ensure only sustained events areidentified, the stack is first smoothed using a 5Ka runningaverage. A total of 36 events are identified, 20 in the earlyPleistocene and 16 in the late. This large number of events,relative to the seven glacial terminations considered inHW05, permits more accurate differentiation between H0
and H1. If a different magnitude is used as the criterion for
identification of events, for example one-half or twostandard deviations of the stack, the number of identifiedevents changes, but the results of the hypothesis test areunaffected.Two options for defining a unique time for each deglacial
event are the half-way-point in time or the half-way-pointin d18 O between the local minimum and maximumbracketing each deglaciation. Here the half-way-point intime is used because this is independent of the particularshape of the deglacial event. Test results are insensitive towhich definition is used. Fig. 5 shows each deglaciationevent.
3.2. Rayleigh’s R
To measure the relationship between the timing ofdeglacial events and orbital variations it is useful to employRayleigh’s R (see Upton and Fingleton, 1989; HW05).First, the phase of obliquity is sampled at the mid-point ofeach deglacial event. Rayleigh’s R is then calculated byconverting phases into unit vectors and computing thevector average,
R ¼1
N
XN
n¼1
cos fn þ i sin fn
�����
�����. (2)
Here, fn is the phase of obliquity sampled at the nthdeglacial event, and the vertical bars indicate the magni-tude. R is real and non-negative with a maximum value ofone when the phases are all the same.
Fig. 5. Obliquity pacing over the last 2Ma. (a) d18 O stack on the extended depth-derived agemodel. The magnitude of one standard deviation in d18 O is
indicated at right, and deglacial events exceeding this magnitude are indicated by a dot. Horizontal bars indicate the two-standard-deviation agemodel
uncertainty. Intervals where two or more obliquity cycles elapse between deglacial events are shaded. (b) The time variability of Earth’s obliquity in
degrees with the mid-point of each deglacial event indicated by a dot. (c,d) Unit circle with obliquity phases during each deglacial event plotted for the
early (c) and late (d) Pleistocene. The vector average associated with each group of phases (Rayleigh’s R value) exceeds the 99% confidence level indicated
by the shaded circle. The one-standard-deviation uncertainty in mean phase is indicated by the arc. Numbers above the obliquity record and plotted on the
Rayleigh circles count the number of obliquity cycles starting from the present.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–55 43
Compared with the more conventional use of Fourieranalysis, Rayleigh’s R is well-suited for measuring therelationship between orbital and glacial variability. First,nonlinearities associated with the variable duration andasymmetry of the glacial cycles do not affect the statistic.Such nonlinearities complicate the Fourier representation,creating overtones and redistributing spectral energythroughout the continuum. Second, agemodel errors causelinear changes in the phase whereas even small agemodelerrors can cause large and complicated distortions of theFourier spectrum (see Thomson and Robinson, 1996).Finally, compared with most measures of phase-coupling,Rayleigh’s R requires fewer realizations in order toestablish significance (Upton and Fingleton, 1989).
3.3. Probability distributions for H0 and H1
To obtain a PDF for H0 (that deglaciations areindependent of orbital phasing) it is assumed that theorbital phase is uniformly distributed over 0 to 3601 withrespect to the timing of deglaciations. A realization of R forthe early Pleistocene is obtained by sampling 20 phasesfrom the uniform distribution. By binning 104 suchrealization an estimate of the PDF is obtained. Similarly,an estimate of the PDF for the late Pleistocene is obtained
by binning R values computed using 16 random phases. Ifmore complicated distributions are assumed for thephasing of the orbital variations, such as those derivedfrom using surrogate data (Schreiber and Schmitz, 2000) orensemble runs of a model (HW05), the hypothesis testresults are unchanged.The larger number of deglacial events permits more
stringent testing of the orbital hypothesis. As opposed tothe 5% significance level used in HW05, a 1% significancelevel is used here (i.e. 99% confidence that H0 can berejected). The critical value at which H0 can be rejected forobliquity at the 1% level is R ¼ 0:47 for the earlyPleistocene (having 20 events) and R ¼ 0:52 for the latePleistocene (slightly larger because it has only 16 events).As a rule, test statistics will only be reported to onesignificant figure except when additional figures serve tomake a point.The probability distribution for H1, that deglacial events
always occur during the same phase of obliquity, issomewhat more involved to estimate. A Monte Carlotechnique is used (Press et al., 1999) where each deglacialevent is assumed to initiate at a local maximum (zerophase) of obliquity. However, deglaciations will generallynot be observed to occur at maximum obliquity owing toagemodel errors. To simulate this effect deglacial ages are
ARTICLE IN PRESS
0 0.5 10
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Fig. 6. Hypothesis test results. The left column is for the early Pleistocene (2–1Ma) and the right column is for the late Pleistocene (1Ma to the present).
The top row is for obliquity, middle for precession, and bottom for eccentricity. Dashed lines indicated the probability distribution for H0, that
deglaciations are independent of the orbital phase, while the solid lines are the probability distribution for H1, that deglaciations occur at maxima of an
orbital parameter but are sampled subject to agemodel uncertainty. The Rayleigh’s R values calculated from the data are indicated by the vertical lines. H0
is rejected only for obliquity. Furthermore, the obliquity results are consistent with H1.
Table 2
Summary of hypothesis test results
Early Pleistocene (2–1Ma) Late Pleistocene (1–0Ma)
R cv 1% Power Phase R cv 1% Power Phase
Obliquity 0.7 0.5 0.6 �56� 0.8 0.5 1.0 �28�
Precession 0.2 0.5 0.0 �88� 0.0 0.5 0.3 �56�
Eccentricity 0.1 0.5 1.0 �24� 0.4 0.5 1.0 �12�
Columns from left to right are the observed Rayleigh’s R value, the critical value at which the null-hypothesis can be rejected at the 1% level, the power of
the test, and the one-standard-deviation uncertainty associated with the mean phase. Columns are repeated for the early and late Pleistocene. Only the
obliquity R values permit rejection of the null hypothesis. Early-Pleistocene tests have a lower power and greater phase uncertainty owing to greater
agemodel uncertainty.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–5544
perturbed according to the uncertainties estimated forthe stack (see Section 2). The obliquity phase is thensampled at the perturbed ages and used to calculaterealizations of R for the early and late Pleistocene. Bybinning 104 realizations of R, the probability distributionassociated with H1 is estimated. The estimated distribu-tions for H0 and H1 are shown in Fig. 6. Table 2 lists thecritical values at which H0 can be rejected for each orbitalparameter.
The uncertainty associated with the mean obliquityphase during deglaciations is also estimated using MonteCarlo techniques. The mean phase is calculated for each ofthe 104 realizations discussed above, and the standarddeviation of these mean phases gives the expecteduncertainty. Note that much of the agemodel uncertainty
is systematic—associated with magnetic reversal ages andauto-correlation in accumulation rates—thereby increasinguncertainty in the mean phase. Table 2 lists the mean phaseuncertainty for each orbital parameter.For the hypothesis test to be meaningful, the data must
be capable of distinguishing between H0 and H1. Therelevant quantity is known as the power of the test(e.g. Devore, 2000) and is the probability of correctlyrejecting H0 when H1 is true. A low power indicates thateven if deglaciations always occur at the same phaseof obliquity the test is unlikely to discern this relation-ship. Table 2 lists the power of the test for eachorbital parameter. The obliquity test during the latePleistocene and eccentricity test during both the early andlate Pleistocene are definitive, having powers of nearly 1.
The larger agemodel uncertainty during the early Pleisto-cene decreases the associated obliquity power to 0.6, butwhich is still large enough to permit a meaningful test.The power of the precession tests is small becauseagemodel uncertainty approaches half a precession cycle,and it is impossible to determine whether precession pacesdeglaciations.
3.4. Test results
During the early Pleistocene, the stability of theobliquity phase is significant at the 99% level ðR ¼ 0:7Þ,as expected, given that early Pleistocene glacial cycles areknown to have a 40Ka timescale (Pisias and Moore, 1981;Raymo et al., 1989; Ruddiman et al., 1989). Moreinterestingly, late Pleistocene deglacial events haveR ¼ 0:8. Thus, the late Pleistocene 100Ka world showsgreater obliquity phase stability than the 40Ka world. In asense, we are still in the 40Ka world. The Pleistoceneobliquity phase stability is remarkable, with 33 of the 36deglacial events occurring within �90� of maximumobliquity. Apparently, the timing of deglacial eventsthroughout the Pleistocene are controlled by obliquityvariations. Mean phases are consistent with deglaciationsinitiating during maximum obliquity to within onestandard deviation.
When the identical test is applied to precession andeccentricity, neither shows significant phase stability withrespect to early or late Pleistocene deglaciations. Owing tothe large power associated with the eccentricity test, it isclear that eccentricity does not pace the deglacial events.The precession test is inconclusive owing to the smallpower of the test.
The results of the test are insensitive to plausiblereformulations. If a 5%, rather than 1%, significance levelis adopted test results do not change. If deglacial events areidentified using only the benthic or planktic records,test results are unaltered. Results are also unchanged ifthe Pleistocene record is divided into other intervals, aslong as these span numerous obliquity cycles. It is thusconcluded that obliquity paces both the early and latePleistocene glacial variability. Apparently, the well-knownshift in the period of variability during the mid-Pleistocenebelies an underlying consistency in the record, thatdeglacial events almost always occur during times of highobliquity.
3.5. Insolation
Having confirmed that deglaciations occur during timesof increased obliquity, it is useful to investigate theinsolation pattern associated with this orbital configura-tion. Fig. 7 shows the diurnal average insolation contouredagainst latitude and day of the year, as well as theanomalies associated with maxima in obliquity andprecession. Anomalies are calculated by averaging theinsolation pattern at times of local maxima in obliquity (or
precession) over the last 2Ma and then subtracting themean insolation over the entire 2Ma period.During summer months, maxima in obliquity are
associated with anomalies ranging from �2W=m2 in thetropics to þ15W=m2 at high latitudes. Positive anomaliesin the annual average insolation occur at latitudes above401 and range up to 5W=m2. The redistribution ofinsolation caused by obliquity variations is small relativethe mean, but are sustained, persisting for �10Ka. Forcomparison, consider that glacial terminations involveapproximately 5� 1015 Kg of ice melting per year(Fairbanks, 1989). If the ice is assumed to initially be at�20 �C, the energy required for melting is equivalent to1W=m2 distributed over the Earth’s surface above 50N. Thus,from a simple energetics point of view, even a small imbalancein net incoming radiation can account for deglaciations.Precession is often described as a stronger control on
insolation variability than obliquity. Indeed, diurnalaverage insolation anomalies associated with precessionreach up to 30W=m2 at high Northern Hemispherelatitudes, roughly twice that of obliquity. But these high-latitude positive anomalies occur only during May andJune and are compensated by equally large negativeanomalies during August and September. Thus theinfluence of precession on the insolation integrated overthe summer period can be quite small (see Huybers, 2006).Whether the seasonal redistribution of insolation asso-ciated with precession can help trigger a deglaciationremains an open question probably best addressed usingcoupled ice-sheet–climate models.
3.6. Obliquity cycle skipping
So far discussion has focused on the timing of deglacialevents, but it is also useful to consider when deglacialevents do not occur. During the early Pleistocene, deglacialevents generally occur every obliquity cycle, but there areimportant exceptions where an obliquity cycle is skipped,most notably during Marine Isotope Stage 36 at 1.2Ma—an event previously noted as being anomalous in d18 O(Mudelsee and Schulz, 1997) and Chinese Loess records(Heslop et al., 2002). Near 1.6 and 1.8Ma obliquitymaxima are associated with weak deglacial events, alsogiving �80Ka variability. These long glacial cycles areidentifiable in the individual d18 O records (Figs. 1 and 2).Long cycles at 1.6 and 1.2Ma are found in all six of thed18 O records spanning this interval. The 1.8Ma event ismore ambiguous, appearing in only three of the fourbenthic records and not the planktic record.Cycle skipping is more frequent during the late
Pleistocene, where most deglacial events are separated bytwo (80Ka) or three (120Ka) obliquity cycles. But here toothere are exceptions. Near 0.7 and 0.6Ma deglacial eventsoccur more nearly every 40Ka. As opposed to a distincttransition from short- to long-period glacial variability, theoverall impression is of a progression toward increasedobliquity cycle skipping.
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Fig. 7. Incoming insolation at the top of the atmosphere computed using the orbital solution of Berger and Loutre (1992). (a) Insolation in W/m2
contoured as a function of latitude and day of the year and (b) the annual average insolation. Plots (a,b) represent average conditions over the last 2Ma.
(c) Insolation averaged during each maxima of obliquity during the last 2Ma and contoured as an anomaly from average conditions, along with (d) the
anomaly in the annual average. (e) Insolation anomaly during maximum precession (when Earth is closest to the sun during summer solstice), and (f) the
annual average showing negligibly small changes. Negative insolation anomalies are indicated by dotted contours.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–5546
There does not appear to be any systematic relationshipbetween the time when obliquity beats are skipped andthe amplitude of the obliquity cycles. Of the skippedobliquity cycles, half are associated with larger thanaverage amplitude cycles and half with smaller amplitudecycles (see Fig. 4). It thus appears that deglaciationsoccur during increased obliquity, but obliquity cycleskipping arises from internal climatic factors. For theprecession parameter, terminations are seen to initiatewhen eccentricity (and hence precession variability) is nearzero at 0.8 and 0.4Ma as well as for the most recenttermination. This result indicates that large values of theprecession parameters are not required for initiation of adeglaciation.
4. The Pleistocene progression
The �100Ka glacial cycles have generally been viewed asa mode of climate variability distinct from the 40Kavariations (Hays et al., 1976; Maasch and Saltzman, 1990;Imbrie et al., 1992; Tziperman and Gildor, 2003). In theabsence of any change in the external forcing (Pisias andMoore, 1981), the onset of �100Ka variations has beentaken to mark the presence of an internal climatic transition(Shackleton et al., 1988; Ruddiman et al., 1989; Birchfieldand Ghil, 1993; Park and Maasch, 1993; Tiedemann et al.,1994; Bolton and Maasch, 1995; Mudelsee and Schulz,1997). In keeping with this view, modeling studies haveinvoked a transition—or equivalently a bifurcation—to
describe the early and late Pleistocene glacial variability(Maasch and Saltzman, 1990; Matteucci, 1990; Paillard,1998; Tziperman and Gildor, 2003; Ashkenazy andTziperman, 2004). A bifurcation implies the suddenappearance of a qualitatively different mode of variabilityfor a nonlinear system (e.g. Strogatz, 1994).
There are three lines of evidence suggesting that a singlebifurcation does not adequately describe events in andaround the mid-Pleistocene. The first argument followsfrom the presence of 80Ka glacial cycles prior to the mid-Pleistocene at 1.6 and 1.2Ma and 40Ka glacial cycles aslate as 0.6Ma. Because long-period glacial cycles arepresent during the early Pleistocene, there can be no singletransition to long-period variability. One can invoke thepresence of multiple bifurcations, or the influence ofstochastic processes causing temporary bifurcations, butthe explanatory power of such a description is limited—what series of event could not be described as a set ofbifurcations?
A second line of evidence has to do with regional climateshifts. If the climate system underwent a bifurcation withrespect to glacial variability, it seems that other elements ofthe system would show a contemporaneous shift. Con-versely, gradual changes in the other components orasynchronous shifts are more difficult to rationalize asowing to a single bifurcation.
Ravelo et al. (2004) examined climate records from highlatitudes, the subtropics, and tropics and concludedthat the onset of glacial variability �2:7Ma does notcoincide with a reorganization at low latitudes. A similarconclusion appears to hold for the mid-Pleistocene,near 0.8Ma. Slow cooling of the deep oceans appearsto have completed prior to 1Ma (Billups, 1998; McIntyreet al., 1999; Marlow et al., 2000). The d13 C of NorthAtlantic intermediate depth water, indicative of nutrientcycling and ocean transport, appears to be stable overthe Pleistocene (Raymo et al., 2004). At lower latitudes,d13 C are reported to become more negative near 1Ma,but then return to early-Pleistocene value by 0.4Ma(Raymo et al., 1997). African aridity seems to increasegradually with a transition, if anywhere, near 1.4Ma(DeMenocal, 1995). Chinese loess deposits show a transi-tion to greater variability in mean grain size at 1Ma andin magnetic susceptibility at 0.6Ma (Heslop et al.,2002). Both seasonal upwelling along the California marginand the East–West d18 O gradient across the tropicalPacific shows an increase near 1.7Ma (Ravelo et al.,2004). The Western Equatorial Pacific has approximatelystable average surface temperatures over the last 5Myr(Wara et al., 2005), with an increase in the period andamplitude of variability at 0.5Ma (Medina-Elizalde andLea, 2005). The Eastern Equatorial Pacific shows a coolingtrend over the last 1.5Ma (Liu and Herbert, 2004; Waraet al., 2005).
As is true for nearly all geophysical measurements,Pleistocene climate records show variability at all times andtimescales. Transitions and changes in different component
of the climate system occur continuously over the last2Ma, and the mid-Pleistocene does not appear especiallyanomalous. Also note that the phasing of EquatorialPacific surface temperatures relative to ice-volume varia-tions appears to be stable over the entire Pleistocene(Medina-Elizalde and Lea, 2005), indicating an invariantrelationship between high- and low-latitude climate varia-bility.The final line of evidence for a gradual rather than
sudden transition relies upon the evolution of the statisticalproperties of the d18 O stack. The analysis of the stack iscomplementary to the examination of numerous regionalclimate records in that the stack reflects aggregate changesin both ice volume and temperature. Ice-volume variationsare nearly anti-phased with temperature variations in thetropics (Liu and Herbert, 2004; Medina-Elizalde and Lea,2005) and at high latitudes (Blunier et al., 1998). This anti-phased behavior is conveniently monitored by the d18 O offoraminiferal calcite because greater ice volume and lowertemperatures both serve to increase d18 O values. Further-more, because the agemodel is not orbitally tuned, thestack permits analysis of changes in orbital periodvariability over the last 2Ma without relying on orbitalassumptions.
4.1. Average frequency
Most studies identify the onset of �100Ka variabilitynear 0.8Ma as indicating a transition from one mode ofglacial variability to another (e.g. Shackleton et al., 1988;Ruddiman et al., 1989; Park and Maasch, 1993; Bolton andMaasch, 1995; Mudelsee and Schulz, 1997). The obliquitypacing results, however, indicate that the �100Kavariability is not a pure mode, but is rather derived fromthe skipping of obliquity beats. Thus, focusing on the�100Ka band to the exclusion of other frequencies is toonarrow a definition to accurately quantify Pleistocenetrends in glaciation.An analogy can be made with the sirens of a passing fire
truck. Owing to Doppler shifting of the sound waves, thesirens will sound at increasingly low frequencies. If thepresence of the fire truck was only gauged by monitoring asingle frequency, one might wrongly conclude that itappeared from nowhere.A quantity better able to describe the evolution of the
glacial cycle frequency is the first moment of the spectrum,M1 ¼ P�1 �
PNi¼1pi � si. Here, pi is the power density at
the ith frequency band associated with a central frequencysi, and P is the sum of the power at all the bands. Thequantity M1 indicates the average frequency of thevariability. Only frequencies below 1
15Ka are considered
as the higher frequency variability is damped by smoothingand averaging of the d18 O records.The evolution of statistical quantities is estimated using
a 200Ka rectangular sliding window. Conclusions areunchanged if a longer window is used; a longer windowmakes results less noisy but provides fewer independent
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Fig. 8. The progression of Pleistocene glacial variability. (a) The average d18 O record oriented so that up corresponds to more ice volume. (b) The first
moment of the power spectrum (i.e. the weighted average of the frequency, M1), (c) the mean value and (d) the standard deviation of the d18 O record. (e)
The time derivative of the d18 O record in %Ka�1 and the associated (f) standard deviation and (g) skewness. Skewness of the rate of change in d18 Oindicates the asymmetry between rates of glaciation and deglaciation. The dashed line indicates the least-squares best fit to each trend. Pleistocene glacial
variability is better described by a trend, or progression, than by any single transition. Statistics are computed using a 200Ka sliding window, and
independent realizations of each statistic are indicated by the vertical dotted lines.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–5548
realizations and would broaden any abrupt features.Shorter windows introduce serious aliasing of the�100Ka glacial cycles and thus are not easily interpreted.
Fig. 8b shows that M1 follows an approximately lineartrend, beginning at the 1
40Ka frequency 2Ma and evolving
to a lower 170Ka frequency. Such a trend is evident in the
evolutionary spectrum (Fig. 4) and agrees with theincreasing number of skipped obliquity cycles (Fig. 5).Were this trend to continue, glacial cycles lasting 160Kawould presumably appear.
4.2. Mean and variance
Long-term trends are also evident in the average d18 O,associated with increased ice volume (Raymo, 1994) andice-volume variability. The d18 O maximum near 0.9Ma(stage 22), the largest seen up to that time, has generallybeen identified with a glacial transition (Prell, 1982;Maasch, 1988; De Blonde and Peltier, 1991; Berger andJansen, 1996; Mudelsee and Schulz, 1997). But it can alsobe argued that the glacial maxima at 1.2Ma (stage 36) and
possibly at 0.6Ma (stage 16) represent unprecedentedincreases in d18 O which are associated with transitionstoward longer period variability.
Rather than speaking of a series of independenttransitions, it is probably more useful to describe theseevents as a trend. Fig. 8c shows the mean computed using asliding 200Ka window, giving 10 independent realizationsof the mean value. Changes in the mean value show themost step-like transition of any quantity investigated, buteven here a linear trend is the best order-one description.The residual variance after removing the best-fit trend fromthe 10 independent realizations of the mean is ð0:076%Þ2,somewhat smaller than if the best-fit step wise transition isremoved ð0:082%Þ2.
Mudelsee and Schulz (1997) note that changes in themean value preceded the onset of �100Ka variability, anddescribe this as an unexplained feature of the Pleistocenevariability. Here, a simple explanation is offered, thatincreases in the mean d18 O value do occur concomitantwith increases in the period, but that the identification andcomparison of particular transition times is an inappropri-ate description of the variability.
The standard deviation shows a more linear trend thanthe mean (Fig. 8d). The trend toward lower frequencyvariability and a greater standard deviation may have asimple relationship to one another. For example, in thesimple linear system dx=dt ¼ sinwt the amplitude of theoscillations in x are inversely proportional to thefrequency, w. The amplitude of the derivative of x,however, is insensitive to the forcing period, making itinteresting to explore the time rate of change in d18 O(Fig. 8e). Fig. 8f shows that a trend toward increasingstandard deviation exists even in the time rate of change ind18 O. This indicates that the trend toward lower frequen-cies is not alone sufficient to explain the greater amplitudeof variability. The implication is that the climatic sensitivityto external forcing and/or internal variability has increasedthrough time.
Ravelo et al. (2004) have considered the sensitivity ofglacial variability to obliquity forcing. Sensitivity appears toincrease between 4 and 1Ma, but then declines toward thepresent, seemingly at odds with the above interpretation ofcontinuous obliquity pacing and a continuous increase insensitivity. The difference in interpretation arises because ofthe narrow versus broad-band analysis of the variability.Ravelo et al. (2004) assume a linear relationship between theobliquity forcing and the climatic response. Similarly,Lisiecki and Raymo (in press) interpret the climate responseto orbital forcing within the context of a linear relationship.Here, it is argued that the response to obliquity becameincreasingly nonlinear, resulting in greater variability atperiods other than 40Ka. If variations with 40, 80, and120Ka periods are all considered as related to obliquity, apositive trend is obtained in sensitivity over the last 2Myr,similar to the trend in standard deviation (Fig. 8d).
Note that the interpretation of glacial variability as theforced response to insolation variability is but one
possibility. Another possibility is that glacial cycles are afree mode of variability but which is phase-locked by theinsolation forcing (Saltzman et al., 1984; Tziperman et al.,in press). In this case, the timing of glacial variability maybe controlled by obliquity but the amplitude of thevariability would be largely independent.
4.3. Asymmetry
A final quantity of interest is the skewness of the rate ofchange in d18 O, a measure of the asymmetry betweenaccumulation and ablation. Over the course of thePleistocene a nearly linear trend is present from zeroskewness to increasingly large values (Fig. 8g). Thisindicates that deglaciations became increasingly rapidrelative to ice-sheet growth. The presence of asymmetryin the glacial cycles agrees with the results of Raymo (1992)and Ashkenazy and Tziperman (2004). The existence of atrend in the asymmetry is also consistent with finding ofLisiecki and Raymo (in press).One complexity arises in that sediment composition or
accumulation rates may covary with other climate changes(Herbert, 1994). For instance, if the average rate ofaccumulation was to decrease during deglaciation, ageestimates would be compressed with deglaciations appear-ing more rapid and glacial cycles more asymmetric. Themost recent glacial cycles are known to be highlyasymmetric because the rate of climate change can befound using radiometric techniques (e.g. Thompson andGoldstein, 2005) or annual layer counts (e.g. Meese et al.,1997). These climate-independent dating techniques, how-ever, are not applicable beyond a few hundred-thousandyears ago, making inferences regarding the more subtleearly-Pleistocene asymmetry more circumspect.Taking the observations at face value, asymmetric
variability indicates that a purely linear response to theMilankovitch forcing is an insufficient explanation of theearly-Pleistocene glacial variability. Numerous explana-tions have been put forward for the asymmetry betweenrates of glaciation and deglaciation including the interac-tion between accumulation and isostatic rebound (Le-Treut and Ghil, 1983), ice-sheet instabilities (Pollard, 1983;Marshall and Clark, 2002), decreases in the albedo of agingsnow (Gallee et al., 1992), and changes in accumulationcaused by rapid expansion of sea ice (Tziperman andGildor, 2003). There appears no fundamental reason whyany of these physical mechanisms could not evolvegradually. For example, while the extent of sea ice in theNorth Atlantic sector does change rapidly on a seasonalbasis, a long-term cooling trend could serve to graduallyincrease the amplitude of these changes.Lisiecki and Raymo (in press) have examined trends in a
different compilation of d18 O records (Lisiecki andRaymo, 2005), and come to similar conclusions that thereexist gradual trends toward greater ice volume, variance,and skewness through time and that the mid-Pleistocene isnot marked by any distinct transition. Taken together, the
gradual trend in glacial cycle variability and continuousobliquity pacing indicate that Pleistocene glacial variabilityis better described by a progression than by any singletransition.
5. A simple model
To describe the Pleistocene progression in glacialvariability the simple model of the data presented inHW05 is extended to include a temporal trend,
Vt ¼ Vt�1 þ Zt and if V tXTt terminate,
Tt ¼ atþ b� cy0t. ð3Þ
Here V is ice volume in normalized units, t is time, Zt
represents the balance of accumulation and ablation overone time step, and Tt is a time-variable thresholdconsisting of a linear trend modulated by obliquity, y0.The prime indicates that obliquity is normalized to zeromean and unit variance. Ice volume is parameterized toaccumulate through time until the threshold condition iscrossed, invoking a termination which linearly resets icevolume to zero over 10Ka. The model describes a simplelimit cycle consisting of steady accumulation followed byrapid collapse, a behavior which has been produced by
Fig. 9. Results from the simple model given in Eq. (3). (a) Model output is p
threshold having a linear trend with superimposed obliquity variability. Vertic
maximum obliquity ðR ¼ 0:9Þ. After tuning the model’s three adjustable param
is an evolutionary spectral estimate of the model results. Shading indicates the
(Ka) and frequency (1/Ka). Horizontal lines are at 1100; 141, and 1
22Ka. Similar to
obliquity period during the late Pleistocene, but with increasing obliquity
concentrations of power appear at 1100
and 122Ka bands. Horizontal dashed lin
more sophisticated ice-sheet models (Marshall and Clark,2002). The obliquity modulation of the threshold conditionmay be rationalized in that increased annual average high-latitude insolation could heat an ice-sheet, increasingmelting and lubrication of the ice-sheet base, and increas-ing the likelihood of a collapse (HW05).There are only three adjustable parameters associated
with Eq. (3), each associated with the threshold condition:the slope (a), intercept (b), and obliquity amplitude (c). Netaccumulation is set to Z ¼ 1; adjustments in this parametercan equivalently be made by changing the thresholdconditions. Unlike the model presented in HW05, thismodel is insensitive to initial conditions because thetermination condition always resets ice volume to zeronear the beginning of the model run. Selecting a slope ofa ¼ 0:05Ka�1, an intercept of b ¼ 126, and an obliquityamplitude of c ¼ 20 reproduces the timing of mostdeglaciations over the last 2Ma (see Fig. 9a). Exceptionsare that a deglaciation near 1.35Ma is missed, the longglacial cycle at 1.6Ma is not reproduced, termination3 initiates 10Ka too early, and some of the smallerlate-Pleistocene deglaciations are not reproduced. Oneother shortcoming is that while the amplitudes of thelate-Pleistocene deglaciation are reproduced, the early-Pleistocene variations are too small. The inclusion of a
lotted against the d18 O stack on the depth-derived agemodel. (b) Model
al lines indicate the mid-point of each termination and always occur near
eters (see text), it describes the timing of most deglaciations. (c) At bottom
logarithm of the power density (in units2/cycle/Ka) as a function of time
the spectrogram of the d18 O stack (Fig. 4), energy is concentrated at the
beat skipping and asymmetry between glaciation and ablation rates,
more complicated threshold condition or additionalmodulation of the accumulation rate by orbital variationscould improve the model fit with the observations, but atthe expense of making the model more complicated.
One insight obtained from Eq. (3) pertains to why themid-point of each termination occurs near maximumobliquity. The initiation of a model termination generallyoccurs while obliquity is large but still increasing, i.e.before maximum obliquity. Because the duration of atermination is �10Ka, the termination mid-point occurs5Ka after its initiation, typically in the vicinity ofmaximum obliquity. The mean phase of obliquity duringdeglaciations is 6� 23�, and the Rayleigh’s R value is 0.9,indicating a high degree of phase stability.
An evolutionary spectrum of the model output (Fig. 9c)shows that it reproduces the main spectral featuresassociated with the d18 O record (Fig. 4). Spectral energyremains nearly constant at the 40Ka period. (Moreprecisely, the obliquity period is at 41Ka.) Growth ofenergy is seen at periods longer than obliquity near 1.4Maand culminates in strong �100Ka power by 0.5Ma.Energy at �22Ka periods appears near 1Ma and remainsup to the present. Importantly, this change in modelspectral characteristics occurs without any sudden changein the mode of glacial variability. The gradual increase inthe threshold value causes glacial cycles to more often skipobliquity beats and accounts for the increase in low-frequency variability. Likewise, the increasing thresholdvalue causes the deglaciation to become increasingly rapidand increases the asymmetry of the glacial cycles. Suchasymmetry in a time-series introduces overtones andharmonics in the Fourier spectrum (e.g. Bracewell, 2000)and accounts for the appearance of an overtone of theobliquity period near 2
40¼ 1
20Ka and a combination tone at
1100þ 1
40� 1
29Ka. Note that the concentration of variability at
the 29Ka period in the model output is not found in thed18 O stack, but has been identified in other studies of d18 Ovariability (Yiou et al., 1991; Bolton and Maasch, 1995;Mix et al., 1995b; HW04).
The addition of a stochastic component to the modelsimulates the presence of weather at the highest frequenciesand the myriad climatic processes not resolved by the modelat longer periods (see Wunsch, 2004). Here a stochasticcomponent is parameterized by changing the accumulationterm, Zt, in Eq. (3) from a constant to a random realizationfrom a normal distribution with a mean and standarddeviation of one. The timing of deglaciation is still controlledby obliquity (Rayleigh’s R averages 0:75� 0:2), but obliquitycycle skipping is now random so that the glacial sequenceneed not coincide with the d18 O stack.
Even with the stochastic forcing, the model reproducesthe progression in statistical quantities described inSection 4. By selecting a positive slope for the thresholdvalue, ice volume and its variance will increase throughtime. The fixed rate of accumulation coupled with anincreasing threshold makes the average glacial cyclefrequency decrease. Furthermore, because ice volume is
always made to decrease to zero in a 10Ka period, theasymmetry between rapid deglaciation and slow accumula-tion will increase through time, in agreement with the trendtoward greater skewness in the rate of change of d18 O.A typical realization of the stochastic model and theevolution of its period, mean, variability, and skewness areshown in Fig. 10. Trends are very similar to those observedfor the stack and occur for a wide range of parameteriza-tions and noise conditions.A number of other simple models have been used to
describe Pleistocene glacial variability (e.g. Paillard, 1998;Clark et al., 1999; Tziperman and Gildor, 2003). Each ofthese models relates the �100Ka variability to theprecession forcing. A troublesome feature of these modelsis that the early-Pleistocene variability shows significant�22Ka precession period variability, at odds with the d18 Odata. This suggests that a model relying upon precession togenerate the �100Ka period will, in general, have difficultygenerating predominantly 40Ka period variability duringthe early Pleistocene. Models relying upon 40Ka variationsto pace the �100Ka glacial cycles should more readilyreproduce the early-Pleistocene 40Ka variations.The study by Ashkenazy and Tziperman (2004) com-
pared their model results against the d18 O proxy of glacialvariability. The model achieves a maximum cross-correla-tion with the d18 O record of 0.3 when the d18 O agemodel isnot tuned to orbital variability and 0.5 when the agemodelis tuned. Their model has eight adjustable parametersincluding a switch near 0.9Ma. By comparison, the modelgiven in Eq. (3) has only three adjustable parameters andachieves a cross-correlation of 0.7 with the d18 O stack.That Eq. (3) has fewer adjustable parameters and obtains ahigher cross-correlation indicates a more skillful descrip-tion of the Pleistocene glacial variability.The simplicity of Eq. (3), however, is such that the long-
term trend in the threshold value could arguably beidentified with a number of independent processes.Candidates are a long-term decrease in greenhouse gases(Raymo, 1997) causing global cooling and the ability tosustain larger ice-sheets. A related possibility is that globalcooling effects deep ocean temperature and sea-icevariability (Tziperman and Gildor, 2003). Another candi-date is scouring of the continental regolith (Clark et al.,1999) causing greater friction between the ice-sheet and itsbed and permitting the accumulation of greater continentalice volume. While rationalizations can be offered to relatethe trend in the threshold to physical mechanisms, theanalysis presented here is incapable of distinguishingbetween mechanisms. A more physical model of the glacialcycles will be required to distinguish the controls on thelong-term evolution of the glacial cycles.
6. Further discussion and conclusions
The Pleistocene has generally been described as havingtwo distinct modes of glacial variability characterized by 40and �100Ka periods of variability. Indeed, the early
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Fig. 10. Statistical evolution of a realization of Eq. (3) using the stochastic accumulation parameterization. Results are similar with those of the d18 Ostack (see Fig. 8). (a) Typical realization of the model output. Units are in normalized ice volume. (b) Weighted average of the frequency (M1, see text), (c)
mean value, and (d) standard deviation of the model ice volume. (e) The rate of change in ice volume and the evolution of the associated (f) standard
deviation and (g) skewness. All statistics generally follow a linear trend as indicated by the dashed line.
P. Huybers / Quaternary Science Reviews 26 (2007) 37–5552
Pleistocene has been called Milankovitch’s other unsolvedmystery (Raymo and Nisancioglu, 2003). The continuousobliquity pacing of deglaciations, however, indicates thatboth the early- and late-Pleistocene glacial cycles derivefrom similar mechanisms and that there is but a singlePleistocene glacial mystery. A physical model capable ofgenerating 40Ka variability during the early Pleistocenewill probably also explain the �100Ka variations of thelate Pleistocene.
Continuous obliquity pacing of the Pleistocene glacialvariability is a more simple hypothesis than those callingupon a new mode of variability to explain the �100Ka
late-Pleistocene variability. The obliquity hypothesis alsoresolves or side steps many of the problems facing theconventional orbital theory of the glacial cycles. Tohighlight this point, the major problems associated withthe orbital theory (following the list of Elkibbi and Rial,2001) are listed and addressed below:
(1)
The �100Ka variations in insolation forcing due toeccentricity changes are too small to directly cause theglacial cycles (Imbrie et al., 1993): The results of thehypothesis test (Section 2) indicate that eccentricitydoes not pace the glacial cycles.
The d18 O data do not show a 413Ka period,corresponding to the most energetic eccentricity bandof variability (Imbrie et al., 1993): Again, the period ofglacial variability is unrelated to eccentricity.
(3)
The mid-Pleistocene transition occurs without a corre-sponding change in the insolation forcing (Pisias andMoore, 1981): Pleistocene glaciation undergoes agradual evolution unrelated to changes in the insolationforcing.
(4)
Glacial cycles vary in duration from �80 to �120Kaduring the last pleistocene (Raymo, 1997; Petit et al.,1999): This supports the obliquity pacing hypothesiswhich calls on glacial cycles to be quantized inmultiples of the basic 40Ka period.
(5)
Spectral peaks exist at frequencies other than those inthe insolation forcing (Nobes et al., 1991; Bolton andMaasch, 1995; HW04): This can be explained by theobliquity cycle skipping and the asymmetry betweenrates of deglaciations and accumulation (see Section4.3). Note that much of the variability resides at non-orbital periods indicating the presence of a significantstochastic contribution (Wunsch, 2004) or a signifi-cantly nonlinear response to the insolation forcing(Huybers and Curry, 2006).
(6)
A final issue, not discussed by Elkibbi and Rial (2001),is that the glacial cycles are symmetric between thehemispheres whereas the seasonal precession forcing isanti-symmetric: This is readily addressed in that theannual average and seasonal insolation anomaliescaused by changes in obliquity are, like the glacialcycles, symmetric between the hemispheres (see Section3.5).
A related issue, touched in item (3), is that the transitionfrom 40 to �100Ka modes of glacial variability hasgenerally been described as rapid relative to the duration ofthe Pleistocene. This view derives from the sudden onset of�100Ka variability. But because the �100Ka variability isnot a fundamentally new mode of variability, a moregeneral description of the variability is required. Examina-tion of the evolution of the mean value, variance, averageperiod, and asymmetry of the glacial cycles all indicate thatthe variability slowly evolves over the course of thePleistocene.
A succinct description of Pleistocene glacial variability iscodified in Eq. (3) using only three adjustable parameters.In words, ice tends to accumulate until some threshold isreached, causing a deglaciation. The threshold level ismodulated by obliquity and exhibits a long-term trend. Themodulation causes terminations to occur near maxima inobliquity while the trend causes an increase in the mean,variance, asymmetry, and period of the ice-volumevariability. The model exhibits a gradual progression inits qualitative behavior rather than a bifurcation. That Eq.(3) reproduces the primary structure of the glacialvariability indicates that obliquity pacing alone is asufficient description of Pleistocene glaciation.
Analysis of the d18 O record provides important cluesand constraints regarding the nature of the glacialvariability, but is insufficient to uniquely determine thecauses of the glacial cycles. Major uncertainties remainregarding the Pleistocene glacial variability, including theorigins of the progression in the glacial cycle variability andthe physical mechanisms responsible for glacial termina-tions. Use of more physical models of the climatevariability and incorporation of a greater variety of climateproxies is essential to furthering our understanding ofPleistocene glaciation.
Acknowledgments
Funding was provided by the NOAA PostdoctoralProgram in Climate and Global Change. Useful commentswere provided by Tom Herbert, Kat Huybers, MaureenRaymo, Martin Tingley, and Carl Wunsch.
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